sound waves - university of california, san diegokeni.ucsd.edu/f14/lec3.pdf · 2013. 10. 15. ·...

Sound Waves

Ken Intriligator’s week 3 lectures, Oct 14, 2013

Ken: “OK,let’s get started..”

Tuesday, October 15, 2013

Sound = pressure waves(!x, t)displacement fluctuation

satisfies 3d wave equation:s(!x, t) (∇2 −1

v2∂2

∂t2)s("x, t) = 0

∇2f(r) =

1

r

∂2

∂r2(rf(r))

r = spherical, radial coordinate

∇2

=∂2

∂x2+

∂2

∂y2+

∂2

∂z2


3d wave eqn. solutions s(!x, t) = A cos(!k · !x − ωt + φ)“Plane wave”

unit vector point in dir. of wave velocity

!k = kn̂ k =2π

λn̂ =

ω = vk

“Spherical wave” s(r, t) =A

rcos(kr − ωt + φ)

ω = vk

Consider first plane waves, in the x direction:

s(x, t) = A cos(kx − ωt + φ)


Derive the wave eqn.:p(x, t) = gauge pressure ptot = p0 + p

p = −B(dV/V ) B= “bulk modulus,” it’s just like the spring constant k in F = -kx.

F/V = (m/V )a −∂p

∂x= B

∂2s

∂x2= ρ0

∂2s

∂t2

vsound =

√

B

ρ0

Fits with D.A.!

p = −B∂s

∂x

(“Incompressible” fluid has B=infinity)

∂2s

∂x2=

1

v2

∂2s

∂t2

dx

A


Examples

vsound =

√

B

ρ0

Water: ρ0 = 10

3kg/m3B = 2.18 × 10

9Pavsound ≈ 1480m/s

vsound ≈ 344m/sAir: vgas =√

γkT

m

We’ll understand gasses later. For this week, ignoreall parts of the chapter mentioning temperature T.

Ignore thisnow, for later.


Ear’s f (freq) rangesvsound ≈ 344m/s

vsound = λ/T = λf

20Hz ≤ f ≤ 20, 000Hz1.7cm ≤ λ ≤ 17.5m

Humans:

Dogs: 40Hz ≤ f ≤ 60, 000Hz

Cats: 60Hz ≤ f ≤ 80, 000Hz

Bats: 10, 000Hz ≤ f ≤ 200, 000Hz

75Hz ≤ f ≤ 150, 000HzDolphin:


Piano


Pressure Wave eqn.∂2s

∂x2=

1

v2

∂2s

∂t2

∂2p

∂x2=

1

v2∂2p

∂t2

p = −B∂s

∂x·B

∂

∂x

Displacement wave equation:

So get same wave eqn for pressure

s(x, t) = A cos(k(x − vt) + φ0)

p(x, t) = BkA sin(k(x − vt) + φ0)

pmax = BkA = (vρ)(Aω)Larger A (louder) or higher frequencygives bigger p, this can hurt your ears!

Multiply both sides of the s wave eqn. by

(Derived a few slides ago.)


Ouch!s(x, t) = A cos(k(x − vt) + φ0)

p(x, t) = BkA sin(k(x − vt) + φ0)

pmax = BkA = (vρ)(Aω)Larger A (louder) or higher frequencygives bigger p, this can hurt your ears!

Largest gauge pressure before damaging your ears:

pmax ≈ 28Pa v ≈ 343m/s ρair ≈ 1.21kg/m3

(Aω)max ≈ 28/(343)(1.21) ≈ 0.067m/s

e.g. f = 103Hz = ω/2π Amax ≈ 1.1× 10−5mSound wave amplitude, at this frequency, should

be less, or you’ll damage your ears.


Intensity I =

Power

AreaLine on top means “average”, sometimes we also use

these brackets < > to denote averaging.

I =dE

dt=

dK

dt+

dU

dt= 2

dK

dtLike the harmonic oscillator, all oscillatingsystems have average K and U are equal.

dK

dt=

1

2

dm

dt

�∂s

∂t

�2=

1

2ρArea

dx

dt

�∂s

∂t

�2

=1

2ρ(Area)vsoundω

2s2max sin2(kx− ωt+ φ)

s = smax cos(kx− ωt+ φ)

I =1

2ρvsoundω

2s2max

(sin2(∗) = 12)

(For point source, s ~ 1/r)

Ipoint source = Ps/4πr2

(Area).


Intensity & sound levelβ ≡ (10dB) log10(I/10−12W/m2)

I = 10−12W/m2 → β = 0 “What?” Too quiet to hear.

Conversation has beta about 60dB. Loud rock concert is about 110dB. Pain threshold is about 120dB.

Human senses (hearing, seeing, touch, smell, taste) detect intensities I on a log scale. Which is good! Can detect over a large I range, from faint to huge.


Waves in pipes!s=0 at

closed endsp (gauge) =0 at open ends

p = −B∂s

∂x

∂s

∂x|open = 0s|closed = 0

So, the wave eqn. BCs are:

and

(Just like with astring at fixed end.)

(Just like with astring at a free end.)


Pipe wave harmonics

λ = 4L/nodd λ = 2L/nOne end open, one end closed

Both open, or both closed

(Drawingwave’s s

displacement.Pressure is

~ derivative,so opposite.)


s & p in pipes


http://www.google.com/imgres?q=wave+in+pipe&client=safari&rls=en&biw=1039&bih=635&tbm=isch&tbnid=4yOfGhJDUL8_xM:&imgrefurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html&docid=rQL8NLCVyxArsM&imgurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/Opentubeoneend.jpg&w=1600&h=742&ei=VYNZUoyAK4aOigLhvoCABA&zoom=1&ved=1t:3588,r:26,s:0,i:165&iact=rc&page=3&tbnh=153&tbnw=330&start=26&ndsp=16&tx=209&ty=74http://www.google.com/imgres?q=wave+in+pipe&client=safari&rls=en&biw=1039&bih=635&tbm=isch&tbnid=4yOfGhJDUL8_xM:&imgrefurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html&docid=rQL8NLCVyxArsM&imgurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/Opentubeoneend.jpg&w=1600&h=742&ei=VYNZUoyAK4aOigLhvoCABA&zoom=1&ved=1t:3588,r:26,s:0,i:165&iact=rc&page=3&tbnh=153&tbnw=330&start=26&ndsp=16&tx=209&ty=74http://www.google.com/imgres?q=wave+in+pipe&client=safari&rls=en&biw=1039&bih=635&tbm=isch&tbnid=4yOfGhJDUL8_xM:&imgrefurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html&docid=rQL8NLCVyxArsM&imgurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/Opentubeoneend.jpg&w=1600&h=742&ei=VYNZUoyAK4aOigLhvoCABA&zoom=1&ved=1t:3588,r:26,s:0,i:165&iact=rc&page=3&tbnh=153&tbnw=330&start=26&ndsp=16&tx=209&ty=74http://www.google.com/imgres?q=wave+in+pipe&client=safari&rls=en&biw=1039&bih=635&tbm=isch&tbnid=4yOfGhJDUL8_xM:&imgrefurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html&docid=rQL8NLCVyxArsM&imgurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/Opentubeoneend.jpg&w=1600&h=742&ei=VYNZUoyAK4aOigLhvoCABA&zoom=1&ved=1t:3588,r:26,s:0,i:165&iact=rc&page=3&tbnh=153&tbnw=330&start=26&ndsp=16&tx=209&ty=74http://www.google.com/imgres?q=wave+in+pipe&client=safari&rls=en&biw=1039&bih=635&tbm=isch&tbnid=4yOfGhJDUL8_xM:&imgrefurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html&docid=rQL8NLCVyxArsM&imgurl=http://www.acs.psu.edu/drussell/Demos/StandingWaves/Opentubeoneend.jpg&w=1600&h=742&ei=VYNZUoyAK4aOigLhvoCABA&zoom=1&ved=1t:3588,r:26,s:0,i:165&iact=rc&page=3&tbnh=153&tbnw=330&start=26&ndsp=16&tx=209&ty=74

The math for pipe wave

s(t, x = 0) = 0∂s

∂x(t, x = L) = 0

xz

y x=0 x=L

s = smax sin(kx) cos(kvt+ φ0)

cos(kL) = 0B.C.@L: kL = (n+1

2)π

k = 2π/λ L = (2n+ 1)λ/4


Math for other cases

Both ends closed: s = smax sin(kx) cos(kvt+ φ)

kL = nπB.C.@L: λ = 2L/n

Both ends open:

kL = nπB.C.@L: λ = 2L/n

s = smax cos(kx) cos(kvt+ φ)


FAQ: Why the sound?

fn = nf1 = nvsound/λ1

λ1 = 2L, n = 1, 2, 3, 4 . . .OO or CC:

λ1 = 4L, n = 1, 3, 5, 7 . . .OC:

From last time, get:

here odd only


Example: singing tube

L (e.g. = 1m)hotair

f1 = v/λ1 = v/2L = 344ms−1/2m = 172Hz.


How does it sound?233Hz = vsound/4L


Interference

s = A1 cos(k(L1 − vt) + φ1) +A2 cos(k(L2 − vt) + φ2)

Superpose two traveling waves..

.Two

sources(speakers)

. sounddetector(your ear)

cos a+ cos b = 2 cos(a− b2

) cos(a+ b

2)use

A1 = A2 φ1 = φ2Let’s keep

it simple, take:

s = 2A cos k(L2 − L1

2) cos(k(

L1 + L22

)− ωt+ φ)get

Constructive: k∆L = 2πnk∆L = (2n+ 1)πDestructive:

∆L = nλ

∆L = (n+1

2)λ

Can be louder or quieter(!) than with just one speaker, depending

on your location. Here’s why:

i.e.i.e.


Beats

A cos(ω1t) +A cos(ω2t) = 2A cos(1

2(ω1 − ω2)t) cos(

1

2(ω1 + ω2)t)

Superpose two sources with different frequencies:

Over time, sometimes addconstructively, sometimesdestructively. Hear beats.

2A cos(1

2(ω1 − ω2)t) modulating wave envelope.

Inside, average frequency.


Doppler effect

You’ve all noticed itwhen a moving siren passes you, frequency goes from high to low.

Police radar speed traps also use Doppler’s

effect. So do bats and dolphins and whales,

with their sonar. Tuesday, October 15, 2013

Doppler effect Case 1: vsource = 0, vobserver �= 0

Both source and observer see same wavelength, but differing effective speed of sounds, because the observer sees sound’s velocity relative to their own. So they see sound as

moving slower if they’re moving away from it, or faster if they’re moving toward it.

Case 2:

Wavelength is stretched or compressed, depending on whether the source is moving away ortoward the observer. Put these two cases together to get the general case (next slide).

vsource �= 0, vobserver = 0

λemit = vsound/femit = λobs = (vsound − vobserver)/fobs

λobs = λemit − (vsource/femit) = (vsound − vsource)/femit


Doppler, cont. fobserved =

vsound − vobservervsound − vsource

femitted

vsound − vobservervsound − vsource

≈ 1− vobserver − vsourcevsound

vsound � vo, vsif

then

Aside (later), for light: fobs = fsource�

1± vrel/c1∓ vrel/c

Top sign if source is coming at us (blueshift), bottom sign if it’s moving away (red).

Put together:

Signs: replace - signs with +, depending on v directions.


Sonic boom!

sin θ = vsound/vsource

Mach number = vobject/vsound

= 1/Mach number

If >1.

(Concorde, mach 2around 1350 mph!)


sound waves - university of california, san diegokeni.ucsd.edu/f14/lec3.pdf · 2013. 10. 15. ·...

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